The present invention is directed to a method and apparatus for optical chemical sensing and more particularly, to methods and systems for determining analyte contents of media by measuring spectral properties of a dye materials exposed to the analyte and processing the spectral properties in accordance with a family of ratiometric expressions.
Various forms of analyte-measuring instruments have been designed and developed for use in biotechnology, industrial and environmental applications. Among these devices are those which rely on spectral properties of dye solutions responsive to a particular analyte. When a dye solution interacts with the analyte, the spectral properties of the dye solution change to a degree related to the concentration of the analyte in the surrounding medium. Thus, one may determine the analyte content in an analyte-containing medium by measuring the change in the spectral properties of the dye solution after the dye solution is exposed to the medium.
Two spectral properties of materials which have been used in optical chemical measurements are optical absorption and fluorescence. Absorption occurs when a material is illuminated and a portion of the illuminating light is neither transmitted through the material, scattered by it nor reflected from it. In the case of a layer of a solution, Beer""s Law states that the amount of light energy thus absorbed by a solute in a solution will depend on the wavelength of the illuminating light, the concentration of the solute in the solution and the thickness of the solution layer:
A=xcex5bC,
where xe2x80x9cAxe2x80x9d is the xe2x80x9cabsorbance,xe2x80x9d that is, the common logarithm of the ratio of the intensity of the illuminating light to the intensity of the absorbed light; xe2x80x9cxcex5xe2x80x9d is the xe2x80x9cmolar absorptivityxe2x80x9d of the solute; xe2x80x9cbxe2x80x9d is the thickness of the layer of the solution; and xe2x80x9cCxe2x80x9d is the concentration of the solute in the solution. Typically, the molar absorptivity will vary with the wavelength of the illuminating light, reaching maximum values at a so-called xe2x80x9cpeak absorption wavelengthsxe2x80x9d of the solute.
Fluorescence occurs when light energy is absorbed by the material and subsequently emitted as light energy of a different wavelength. In the case of a thin layer of a dilute aqueous solution, the intensity xe2x80x9cFxe2x80x9d of the fluorescent emission can be expressed as follows:
F=2.3PKxcex5bC,
where xe2x80x9cPxe2x80x9d is the intensity of the illuminating light and K is the xe2x80x9cquantum efficiencyxe2x80x9d of the solute. Like the molar absorptivity xe2x80x9cxcex5,xe2x80x9d the quantum efficiency will vary with the wavelength of the emitted light.
Certain prior art devices included sensor elements or probes in which aqueous dye solutions were trapped between analyte-permeable surfaces and distal ends of optical fibers. When the probes were exposed to analyte-containing media, analyte molecules diffused into the dye solutions. The dye solutions were illuminated at one or more selected wavelengths. The optical fibers gathered light returning from the dye solutions and conducted the light to one or more transducers. These transducers measured the intensities of the light to determine spectral properties of the dye solution. The spectral properties of the dye solutions were then used to determine the analyte content of the media.
Peterson et al. U.S. Pat. No. 4,200,110 proposed a fiber optic pH sensor using phenol red dye copolymerized with an acrylic base polymer to form microspheres. After the microspheres were exposed to a sample to be measured, they were illuminated sequentially by light at the peak excitation wavelength of the conjugate base (that is, a wavelength of illuminating light which maximizes the emitted intensity), given as 560 nm, and at an isosbestic wavelength (that is, a wavelength at which the optical absorbance of the dye microspheres was independent of the pH), given as 485 nm.
Peterson et al. taught that the pH of the sample could be determined from the intensity of the fluorescent emission of the microspheres measured after the microspheres were illuminated at the peak excitation wavelength of the conjugate base of the dye. In addition, they suggested normalizing this measured intensity by dividing it by the intensity of the fluorescent emission measured after the microspheres were illuminated at the isosbestic wavelength. FIG. 3 of Peterson et al. suggested that the correlation of the pH to this normalized intensity was approximately linear over a range from about pH 6.5 to about pH 7.5.
Seitz et al. U.S. Pat. No. 4,548,907 proposed a fluorescence-based optical sensor including 8-hydroxy-1,3,6-pyrenetrisulfonic acid [hereinafter xe2x80x9cHPTSxe2x80x9d] dye immobilized on an ion exchange membrane. After the immobilized HPTS dye was exposed to a sample to be measured, it was illuminated sequentially by monochromatic light at peak excitation wavelengths of the undissociated HPTS dye, given as 405 nm, and of its conjugate base [hereinafter xe2x80x9cPTSxe2x88x92xe2x80x9d], given as 470 nm. The intensity of the fluorescent output of the dye when the dye was illuminated at each wavelength was measured using a photodetector having a narrow sensitivity band centered on 510 nm. A ratio was calculated by dividing the intensity of the fluorescent output of the dye after the dye was illuminated at the excitation wavelength of PTSxe2x88x92, that is, at 470 nm, by the intensity of the fluorescent output of the dye after the dye was illuminated at the excitation wavelength of HPTS, that is, at 405 nm.
Seitz et al. taught that the correlation between the pH of the sample and the foregoing ratio could be treated as approximately linear over a range from about pH 6 to about pH 8. They recommended using this ratio to determine pH instead of the absolute value of a single measured intensity because the ratio was insensitive to factors such as source intensity variations, fluorescence quenching and degradation of the dye material.
Seitz et al. also proposed the use of their fluorescence-based optical sensor for measuring dissolved carbon dioxide in accordance with the so-called xe2x80x9cSeveringhousexe2x80x9d method. More particularly, they proposed measuring dissolved carbon dioxide in a sample by exposing the HPTS to a solution of sodium bicarbonate of known concentration which, in turn, was exposed to the sample. Carbonic acid ions from the sample tended to lower the pH of the sodium bicarbonate solution. Seitz et al. taught that, by using their optical sensor to measure the pH of the sodium bicarbonate solution, they could determine the concentration of dissolved carbon dioxide in the sample.
Peterson et al. U.S. Pat. No. 4,476,870 proposed a fiber optic probe for determining the partial pressure of oxygen in blood by measuring the fluorescence quenching of a suitable dye exposed to a sample to be measured. More specifically, the dye was exposed to the sample and illuminated with blue light. The intensity of the fluorescent output from the dye, measured in the green range, was correlated to the partial pressure of oxygen by the following formula:             P              O        2              =                            P          xe2x80x2                ⁡                  [                                                    (                                                      I                    blue                                                        I                    green                                                  )                            m                        -            1                    ]                    n        ,
where xe2x80x9cPO2xe2x80x9d was the partial pressure of oxygen; xe2x80x9cIbluexe2x80x9dwas the intensity of the blue illuminating light; xe2x80x9cIgreenxe2x80x9d was the intensity of the green fluorescent emission; xe2x80x9cPxe2x80x2xe2x80x9d was a constant having the dimensions of pressure; and xe2x80x9cmxe2x80x9d and xe2x80x9cnxe2x80x9d were non-dimensional constants.
One particularly demanding application for a chemical sensor is the measurement of the partial pressure of dissolved carbon dioxide in sea water. An oceanographic sensor may be called on to resolve differences in CO2 partial pressure of as little as 1-2 ppm. The sensor must be compatible chemically with sea water and capable of providing accurate measurements despite relatively wide temperature variations.
Dr. David R. Walts of Tufts University proposed an oceanographic sensor in which dissolved CO2 is detected in sea water by means of fluorescent emission intensity measurements performed on an aqueous solution of carboxy-seminapthofluorescein [hereinafter xe2x80x9cc-SNAFLxe2x80x9d] exposed to the sea water through a gas permeable membrane. The c-SNAFL solution was excited by light at a wavelength of 488 nm and the intensities of the fluorescent emissions from the c-SNAFL solution were measured at wavelengths of approximately 545 nm and 625 nm. A non-linear relationship was found to exist between a simple ratio of the fluorescence intensities, F545 nm/F625 nm, and the CO2 partial pressure over a range of about 0-1,000 ppm CO2.
Additional optical chemical sensors were proposed in Furlong U.S. Pat. No. 5,672,515; Walt U.S. Pat. No. 5,252,494; Walt U.S. Pat. No. 5,114,864; Leader et al. U.S. Pat. No. 5,093,266; Matthews et al. U.S. Pat. No. 4,851,195; and Leader et al. U.S. Pat. No. 4,833,091.
Despite such proposals, there remains a need in the art for an optical chemical sensing method and system in which a strong linear correlation between the analyte content and a known function of the spectral properties of the dye solution promotes high resolution over a wide range of analyte content. At the same time, it is desired that the effects of factors, such as illumination intensity variations and photobleaching of the dye material, on the accuracy of the sensor system be minimized.
The present invention is directed to an optical chemical sensing method and system for determining an analyte content of a medium, particularly an aqueous medium. As used herein, the term xe2x80x9canalyte contentxe2x80x9d is not intended to refer to a particular measure (such as concentration or partial pressure) but rather to any quantity denoting a relationship of the analyte to the medium as a whole. Examples of such quantities are the pH of an aqueous solution and the partial pressure of carbon dioxide on a fluid.
More particularly, the method and system of the invention determine the analyte content of a medium as a function of the spectral properties of a dye solution. Preferred dye solutions comprise dyes which dissociate to form conjugate ions having spectral properties different than those of the undissociated dye molecules. Changes in hydrogen ion concentration triggered by exposure of the dye solution to the analyte shift the equilibrium of the dissociation reaction, thereby altering the overall spectral properties of the dye solution. The analyte content may be determined as a function of the hydrogen ion concentration in the vicinity of the dye solution, which, in turn, preferably is determined as a ratio of linear combinations of spectral properties of the dye solution measured at two different wavelengths.
Thus, a preferred method in accordance with the present invention comprises the steps of exposing the dye solution to the analyte; illuminating the dye material to induce a first output light from said dye solution corresponding to a first wavelength and a second output light from said dye solution corresponding to a second wavelength; measuring an intensity of the first output light to determine a first spectral property xe2x80x9cX1;xe2x80x9d measuring an intensity of the second output light to determine a second spectral property xe2x80x9cX2;xe2x80x9d and processing the first and second spectral properties according to an expression of the form:             Y      analyte        =                  f        ⁡                  (                      [                          H              +                        ]                    )                    =              f        ⁡                  (                                                    k                1                            ⁢                              k                2                                      -                                                            X                  1                                -                                                      k                    3                                    ⁢                                      X                    2                                                                                                X                  2                                -                                                      k                    4                                    ⁢                                      X                    1                                                                                )                      ,
where xe2x80x9cYanalytexe2x80x9d is the analyte content of the medium: xe2x80x9cfxe2x80x9d is a function which relates the hydrogen ion content in the vicinity of the dye solution to the analyte content of the medium; xe2x80x9ck1xe2x80x9d is a constant which preferably depends primarily on the chemical properties of the dye; and xe2x80x9ck2,xe2x80x9d xe2x80x9ck3xe2x80x9d and xe2x80x9ck4xe2x80x9d are constants which preferably depend primarily on spectral properties of the undissociated dye molecules and the conjugate ions.
For present purposes, an expression of the form:             X      1        -                  k        3            ⁢              X        2                        X      2        -                  k        4            ⁢              X        1            
will be referred to as a xe2x80x9ccomplex ratio,xe2x80x9d as distinguished from a xe2x80x9csimple ratioxe2x80x9d of the form X1/X2.
Preferred dye solutions for use in the method of the invention include dilute aqueous solutions of weakly acidic dyes such as HPTS and c-SNAFL. Where the dye is a weak acid, the constant k1 is preferably the dissociation constant of the dye.
One especially preferred method in accordance with the invention uses a weakly acidic dye for measuring the pH of an aqueous medium. By definition, pH=xe2x88x92log[H+]. Applying this function of [H+] to the complex ratiometric expression given earlier:       pH    =                  pK        a            -              log        ⁡                  (                                    k              2                        ⁢                          xe2x80x83                        ⁢                                                            X                  1                                -                                                      k                    3                                    ⁢                                      X                    2                                                                                                X                  2                                -                                                      k                    4                                    ⁢                                      X                    1                                                                                )                      ,
where pKaxe2x89xa1xe2x88x92log Ka, xe2x80x9cKaxe2x80x9d being the dissociation constant of the dye.
Another especially preferred method in accordance with the invention uses a dye material comprising an aqueous solution including a weakly acidic dye and an HCO3xe2x88x92 source to measure the partial pressure of carbon dioxide on a medium. In an aqueous solution. dissolved carbon dioxide reacts with water to form carbonic acid, which dissociates into hydrogen ions and bicarbonate:
CO2+H2O⇄H++HCO3xe2x88x92.
The degree to which this reaction occurs at equilibrium will depend on the concentration of dissolved hydrogen ions in the solution:                               [                      H            +                    ]                ⁡                  [                      HCO            3            -                    ]                            [                  CO          2                ]              =          K      a1        ,
where xe2x80x9cKalxe2x80x9d is the reaction constant for carrying out this reaction in aqueous solution. Rearranging this equation:       [          CO      2        ]    =                              [                      HCO            3            -                    ]                          K          a1                    ⁡              [                  H          +                ]              .  
Henry""s law states that the concentration of a gas dissolved in a solution is proportional to the partial pressure of the gas. Applying Henry""s Law to the previous equation yields:       P          CO      2        =                    [                  CO          2                ]                    K        H              =                                        1                                          K                H                            ⁢                              K                a1                                              ⁡                      [                          HCO              3              -                        ]                          ⁡                  [                      H            +                    ]                    .      
where xe2x80x9cPCO2xe2x80x9d is the partial pressure of carbon dioxide and xe2x80x9cKHxe2x80x9d is the Henry""s Law constant. Applying this function of [H+] to the complex ratiometric expression given earlier:       P          CO      2        =                              K          a                                      K            H                    ⁢                      K            a1                              ⁡              [                  HCO          3          -                ]              ⁢                  (                              k            2                    ⁢                                                    X                1                            -                                                k                  3                                ⁢                                  X                  2                                                                                    X                2                            -                                                k                  4                                ⁢                                  X                  1                                                                    )            .      
Measurements performed on test solutions have shown very strong linear correlations between the pH and PCO2 values of the test solutions, on the one hand, and the values derived from complex ratiometric expressions of the form set forth above, on the other. Experiments performed on batch solutions of c-SNAFL and HCO3xe2x88x92 indicate that resolutions of the order of magnitude of parts per million might be obtained over a range of carbon dioxide partial pressures on the order of 0-2,000 ppm. These results suggest that methods within the scope of the present invention are capable of the resolutions necessary for oceanographic CO2 measurements.
A preferred sensor system for use in accordance with the present invention includes a sensor element or probe containing the dye material; at least one light source in optical communication with the probe for illuminating the dye material; at least one frequency-sensitive photodetector for measuring the intensity of light output from the probe to determine a first spectral property xe2x80x9cX1xe2x80x9d of the dye corresponding to a first wavelength and a second spectral property xe2x80x9cX2xe2x80x9d corresponding to a second wavelength; and a controller for processing the first and second spectral properties to determine the analyte content in accordance with an expression of the form:             Y      analyte        =                  f        ⁡                  (                      [                          H              +                        ]                    )                    =              f        ⁡                  (                                    k              1                        ⁢                          k              2                        ⁢                                                            X                  1                                -                                                      k                    3                                    ⁢                                      X                    2                                                                                                X                  2                                -                                                      k                    4                                    ⁢                                      X                    1                                                                                )                      ,
where xe2x80x9cYanalytexe2x80x9d is the analyte content of the medium; xe2x80x9cfxe2x80x9d is a function which relates the hydrogen ion content in the vicinity of the dye solution to the analyte content of the medium; and xe2x80x9ck1,xe2x80x9d xe2x80x9ck2,xe2x80x9d xe2x80x9ck3xe2x80x9d and xe2x80x9ck4xe2x80x9d are constants.
One advantage of the present invention is that it provides a straightforward mode of calibration. The coefficients xe2x80x9ck2,xe2x80x9d xe2x80x9ck3xe2x80x9d and xe2x80x9ck4xe2x80x9d are preferably each ratios of spectral properties of the undissociated dye molecules and the conjugate ions. Thus, a system in accordance with the invention may be calibrated merely by determining the spectral properties of the dye solution corresponding to the first and second wavelengths of light when the dye solution is exposed to buffer solutions of very high pH and very low pH. Since the method of the invention relates the analyte content of the medium to spectral properties of the dye material through a ratiometric expression, it can be anticipated that the system will be relatively insensitive to potential errors due to variations in the intensity of the illuminating light and to photobleaching of the dye solution.
Without wishing to be bound by any theory of operation, it is believed that, where a weakly acidic dye solution is used, the undissociated dye molecules, xe2x80x9cHB,xe2x80x9d reach an equilibrium with the conjugate base ions, xe2x80x9cBxe2x88x92:xe2x80x9d
HB=H++Bxe2x88x92.
The degree to which this dissociation occurs at equilibrium will depend on the concentration of dissolved hydrogen ions in the vicinity of the dye solution:                     [                  H          +                ]            ⁡              [                  B          -                ]                    [      HB      ]        =            K      a        .  
Rearranging this equation:       [          H      +        ]    =            K              a        ⁢                  xe2x80x83                      ⁢                            [          HB          ]                          [                      B            -                    ]                    .      
Note that the sum CHBxe2x89xa1[HB]+[Bxe2x88x92], that is, the total amount of dye added to the solution, remains constant independently of the concentration of dissolved hydrogen ions in the vicinity of the dye solution.
In accordance with one embodiment, analyte content is determined using absorbances of the dye solution measured after a thin layer of the dye solution is exposed to the analyte and illuminated at first and second wavelengths of light. It is assumed that only the undissociated dye molecules and their conjugate base ions contribute to the absorbance of the dye solution. Then, the absorbance of the thin layer of the dye solution will be expressible in accordance with Beer""s Law:
A1=AHB1+ABxe2x88x921=xcex5HB1b[HB]+xcex5Bxe2x88x921b[Bxe2x88x92]
A2=AHB2+ABxe2x88x922=xcex5HB2b[HB]+xcex5Bxe2x88x922b[Bxe2x88x92]
where:
xe2x80x9cA1xe2x80x9d is the absorbance measured after the dye solution is illuminated by the first illumination wavelength;
xe2x80x9cAHB1xe2x80x9d is that part of the absorbance, measured after the dye solution is illuminated by the first illumination wavelength, which is attributable to the undissociated dye molecule in the solution;
xe2x80x9cABxe2x88x921xe2x80x9d is that part of the absorbance, measured after the dye solution is illuminated by the first illumination wavelength, which is attributable to the conjugate base ion in the solution;
xe2x80x9cxcex5HB1xe2x80x9d is the molar absorptivity of the undissociated dye molecule corresponding to the first illumination wavelength;
xe2x80x9cbxe2x80x9d is the thickness of the dye solution layer;
xe2x80x9cxcex5Bxe2x88x921xe2x80x9d is the molar absorptivity of the conjugate base ion corresponding to the first illumination wavelength; and
xe2x80x9cA2,xe2x80x9d xe2x80x9cAHB2,xe2x80x9d xe2x80x9cABxe2x88x922,xe2x80x9d xe2x80x9cxcex5HB2xe2x80x9d and xe2x80x9cxcex5Bxe2x88x922xe2x80x9d are analogous quantities measured with respect to illumination of the dye solution at the second illumination wavelength.
Since the foregoing equations are linear, they can be solved to express the concentrations as linear functions of the measured absorbances as follows:                               [          HB          ]                =                                                                              ε                  2                                      B                    -                                                  /                b                                                                                  ε                    1                    HB                                    ⁢                                      ε                    2                                          B                      -                                                                      -                                                      ε                    2                    HB                                    ⁢                                      ε                    1                                          B                      -                                                                                            ⁢                          A              1                                -                                                                      ε                  1                                      B                    -                                                  /                b                                                                                  ε                    1                    HB                                    ⁢                                      ε                    2                                          B                      -                                                                      -                                                      ε                    2                    HB                                    ⁢                                      ε                    1                                          B                      -                                                                                            ⁢                          A              2                                          ;        ⁢          
        [          B      -        ]    =                                          ε            1            HB                    /          b                                                    ε              1              HB                        ⁢                          ε              2                              B                -                                              -                                    ε              2              HB                        ⁢                          ε              1                              B                -                                                        ⁢              A        2              -                                        ε            2            HB                    /          b                                                    ε              1              HB                        ⁢                          ε              2                              B                -                                              -                                    ε              2              HB                        ⁢                          ε              1                              B                -                                                        ⁢                        A          1                .            
Thus:       [          H      +        ]    =                    K                  a          ⁢                      xe2x80x83                              ⁢                        [          HB          ]                          [                      B            -                    ]                      =                  K        a            ⁢                                                                  ε                2                                  B                  -                                            ⁢                              A                1                                      -                                          ε                1                                  B                  -                                            ⁢                              A                2                                                                                        ε                1                HB                            ⁢                              A                2                                      -                                          ε                2                HB                            ⁢                              A                1                                                    .            
When the dye solution is exposed to a buffer solution having a high pH, all but a negligible amount of the dye will dissociate to form the conjugate base. Under these conditions, the remaining undissociated dye molecules will not absorb a significant amount of light. The absorbance of the dye solution then will take approximately the following form:
(A1)high pH=(ABxe2x88x921)high pH=xcex5Bxe2x88x921bCHB;
(A2)high pH=(ABxe2x88x922)high pH=xcex5Bxe2x88x922bCHB.
Rearranging terms:                     ε        1                  B          -                    =                                    (                          A              1                        )                                high            ⁢                          xe2x80x83                        ⁢            pH                                    bC          HB                      ;              ε      2              B        -              =                                        (                          A              2                        )                                high            ⁢                          xe2x80x83                        ⁢            pH                                    bC          HB                    .      
When the dye solution is exposed to a buffer solution having a low pH, only a negligible amount of the dye will dissociate to form the conjugate base. Under these conditions, the remaining conjugate base ions will not absorb a significant amount of light. The absorbance of the dye solution then will take approximately the following form:
(A1)low pH=(AHB1)low pH=xcex5HB1bCHB;
(A2)low pH=(AHB2)low pH=xcex5HB2bCHB.
Rearranging terms:                     ε        1        HB            =                                    (                          A              1                        )                                low            ⁢                          xe2x80x83                        ⁢            pH                                    bC          HB                      ;              ε      2      HB        =                                        (                          A              2                        )                                low            ⁢                          xe2x80x83                        ⁢            pH                                    bC          HB                    .      
Substituting these values into the expression for the concentration of dissolved hydrogen ions:                               [                      H            +                    ]                =                              K            a                    ⁢                      xe2x80x83                    ⁢                                                                                          (                                          A                      2                                        )                                                        high                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                  ⁢                                  A                  1                                            -                                                                    (                                          A                      1                                        )                                                        high                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                  ⁢                                  A                  2                                                                                                                          (                                          A                      1                                        )                                                        low                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                  ⁢                                  A                  2                                            -                                                                    (                                          A                      2                                        )                                                        low                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                  ⁢                                  A                  1                                                                                            =                                            K              a                        ⁡                          [                                                                    (                                          A                      2                                        )                                                        high                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                                                        (                                          A                      1                                        )                                                        low                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                              ]                                ⁢                      xe2x80x83                    ⁢                                                                      A                  1                                -                                                      [                                                                                            (                                                      A                            1                                                    )                                                                          high                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                                                      (                                                      A                            2                                                    )                                                                          high                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                      ]                                    ⁢                                      A                    2                                                                                                A                  2                                -                                                      [                                                                                            (                                                      A                            2                                                    )                                                                          low                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                                                      (                                                      A                            1                                                    )                                                                          low                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                      ]                                    ⁢                                      A                    1                                                                        .                              
As discussed earlier, the present invention determines analyte content in a medium as a function of hydrogen ion concentration near the dye material. Otherwise stated:       Y    analyte    =            f      ⁡              (                  [                      H            +                    ]                )              =                  f        ⁡                  (                                                    K                a                            ⁡                              [                                                                            (                                              A                        2                                            )                                                              high                      ⁢                                              xe2x80x83                                            ⁢                      pH                                                                                                  (                                              A                        1                                            )                                                              low                      ⁢                                              xe2x80x83                                            ⁢                      pH                                                                      ]                                      ⁢                          xe2x80x83                        ⁢                                                            A                  1                                -                                                      [                                                                                            (                                                      A                            1                                                    )                                                                          high                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                                                      (                                                      A                            2                                                    )                                                                          high                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                      ]                                    ⁢                                      A                    2                                                                                                A                  2                                -                                                      [                                                                                            (                                                      A                            2                                                    )                                                                          low                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                                                      (                                                      A                            1                                                    )                                                                          low                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                      ]                                    ⁢                                      A                    1                                                                                )                    .      
In accordance with another embodiment, analyte content is determined using fluorescent emission intensities of the dye solution corresponding to first and second wavelengths of light. It is assumed that the dye solution is dilute and that only the undissociated dye molecules and their conjugate base ions contribute to the fluorescent emission of the dye solution. Then, the fluorescent emission of a thin layer of the dye solution will be expressible as follows:
F1=FHB1+FBxe2x88x921=2.3P1KHB1xcex5HB1b[HB]+2.3P1KBxe2x88x921xcex5Bxe2x88x921b[Bxe2x88x92];
F2=FHB2+FBxe2x88x922=2.3P2KHB2xcex5HB2b[HB]+2.3P2KBxe2x88x922xcex5Bxe2x88x922b[Bxe2x88x92];
where:
xe2x80x9cF1xe2x80x9d is the intensity of the fluorescent emission corresponding to a first wavelength;
xe2x80x9cFHB1xe2x80x9d is that part of the intensity of the fluorescent emission corresponding to the first wavelength which is attributable to the undissociated acid molecule in the dye solution;
xe2x80x9cFBxe2x88x921xe2x80x9d is that part of the intensity of the fluorescent emission corresponding to the first wavelength which is attributable to the conjugate base ions in the dye solution;
xe2x80x9cP1xe2x80x9d is the intensity of the illuminating light corresponding to the first wavelength;
xe2x80x9cKHB1xe2x80x9d is the quantum efficiency of the undissociated dye molecules corresponding to the first wavelength;
xe2x80x9cKBxe2x88x921xe2x80x9d is the quantum efficiency of the conjugate base ions corresponding to the first wavelength; and
xe2x80x9cF2,xe2x80x9d xe2x80x9cFHB2,xe2x80x9d xe2x80x9cFbxe2x88x922xe2x80x9d xe2x80x9cP2,xe2x80x9d xe2x80x9cKHB2,xe2x80x9d and xe2x80x9cKBxe2x88x922xe2x80x9d are analogous quantities corresponding with a second wavelength.
Since the equations are linear, they can be solved to express the concentrations as linear functions of normalized fluorescent emission intensities (that is, the ratio of the fluorescent emission intensity divided by the illumination intensity) as follows:                               [          HB          ]                =                  xe2x80x83                ⁢                                                                              K                  2                                      B                    -                                                  ⁢                                                      ε                    2                                          B                      -                                                        /                  2.3                                ⁢                                  xe2x80x83                                ⁢                b                                                                                  K                    1                    HB                                    ⁢                                      ε                    1                    HB                                    ⁢                                      K                    2                                          B                      -                                                        ⁢                                      ε                    2                                          B                      -                                                                      -                                                      K                    2                    HB                                    ⁢                                      ε                    2                    HB                                    ⁢                                      K                    1                                          B                      -                                                        ⁢                                      ε                    1                                          B                      -                                                                                            ⁢                          (                                                F                  1                                /                                  P                  1                                            )                                -                                                  xe2x80x83                ⁢                                                                              K                  1                                      B                    -                                                  ⁢                                                      ε                    1                                          B                      -                                                        /                  2.3                                ⁢                                  xe2x80x83                                ⁢                b                                                                                  K                    1                    HB                                    ⁢                                      ε                    1                    HB                                    ⁢                                      K                    2                                          B                      -                                                        ⁢                                      ε                    2                                          B                      -                                                                      -                                                      K                    2                    HB                                    ⁢                                      ε                    2                    HB                                    ⁢                                      K                    1                                          B                      -                                                        ⁢                                      ε                    1                                          B                      -                                                                                            ⁢                          (                                                F                  2                                /                                  P                  2                                            )                                ;                                                  [                      B            -                    ]                =                  xe2x80x83                ⁢                                                                              K                  1                  HB                                ⁢                                                      ε                    1                    HB                                    /                  2.3                                ⁢                                  xe2x80x83                                ⁢                b                                                                                  K                    1                    HB                                    ⁢                                      ε                    1                    HB                                    ⁢                                      K                    2                                          B                      -                                                        ⁢                                      ε                    2                                          B                      -                                                                      -                                                      K                    2                    HB                                    ⁢                                      ε                    2                    HB                                    ⁢                                      K                    1                                          B                      -                                                        ⁢                                      ε                    1                                          B                      -                                                                                            ⁢                          (                                                F                  2                                /                                  P                  2                                            )                                -                                                  xe2x80x83                ⁢                                                            K                2                HB                            ⁢                                                ε                  2                  HB                                /                2.3                            ⁢                              xe2x80x83                            ⁢              b                                                                        K                  1                  HB                                ⁢                                  ε                  1                  HB                                ⁢                                  K                  2                                      B                    -                                                  ⁢                                  ε                  2                                      B                    -                                                              -                                                K                  2                  HB                                ⁢                                  ε                  2                  HB                                ⁢                                  K                  1                                      B                    -                                                  ⁢                                  ε                  1                                      B                    -                                                                                ⁢                                    (                                                F                  1                                /                                  P                  1                                            )                        .                              
Thus:       [          H      +        ]    =                    K        a            ⁢              xe2x80x83            ⁢                        [          HB          ]                          [                      B            -                    ]                      =                  K        a            ⁢              xe2x80x83            ⁢                                                                  K                2                                  B                  -                                            ⁢                              ε                2                                  B                  -                                            ⁢                                                F                  1                                /                                  P                  1                                                      -                                          K                1                                  B                  -                                            ⁢                              ε                1                                  B                  -                                            ⁢                                                F                  2                                /                                  P                  2                                                                                                        K                1                HB                            ⁢                              ε                1                HB                            ⁢                                                F                  2                                /                                  P                  2                                                      -                                          K                2                HB                            ⁢                              ε                2                HB                            ⁢                                                F                  1                                /                                  P                  1                                                                    .            
After calibration with buffer solutions of high pH and low pH, respectively, the products of the molar absorptivitys and the quantum efficiencies will take approximately the form:                               K          1                      B            -                          ⁢                  ε          1                      B            -                              =                                    (                                          F                1                            /                              P                1                                      )                                high            ⁢                          xe2x80x83                        ⁢            pH                                    bC          HB                      ;                                K          2                      B            -                          ⁢                  ε          2                      B            -                              =                                    (                                          F                2                            /                              P                2                                      )                                high            ⁢                          xe2x80x83                        ⁢            pH                                    bC          HB                      ;                                K          1          HB                ⁢                  ε          1          HB                    =                                    (                                          F                1                            /                              P                1                                      )                                low            ⁢                          xe2x80x83                        ⁢            pH                                    bC          HB                      ;                      K        2        HB            ⁢              ε        2        HB              =                                        (                                          F                2                            /                              P                2                                      )                                low            ⁢                          xe2x80x83                        ⁢            pH                                    bC          HB                    .      
Substituting these values into the expression for the concentration of dissolved hydrogen ions:                               [                      H            +                    ]                =                  xe2x80x83                ⁢                              K            a                    ⁢                      xe2x80x83                    ⁢                                                                                          (                                                                  F                        2                                            /                                              P                        2                                                              )                                                        high                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                  ⁢                                                      F                    1                                    /                                      P                    1                                                              -                                                                    (                                                                  F                        1                                            /                                              P                        1                                                              )                                                        high                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                  ⁢                                                      F                    2                                    /                                      P                    2                                                                                                                                            (                                                                  F                        1                                            /                                              P                        1                                                              )                                                        low                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                  ⁢                                                      F                    2                                    /                                      P                    2                                                              -                                                                    (                                                                  F                        2                                            /                                              P                        2                                                              )                                                        low                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                  ⁢                                                      F                    1                                    /                                      P                    1                                                                                                              =                  xe2x80x83                ⁢                                            K              a                        ⁡                          [                                                                    (                                                                  F                        2                                            /                                              P                        2                                                              )                                                        high                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                                                        (                                                                  F                        1                                            /                                              P                        1                                                              )                                                        low                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                              ]                                ⁢                                    (                                                                                          F                      1                                        /                                          P                      1                                                        -                                                            [                                                                                                    (                                                                                          F                                1                                                            /                                                              P                                1                                                                                      )                                                                                high                            ⁢                                                          xe2x80x83                                                        ⁢                            pH                                                                                                                                (                                                                                          F                                2                                                            /                                                              P                                2                                                                                      )                                                                                high                            ⁢                                                          xe2x80x83                                                        ⁢                            pH                                                                                              ]                                        ⁢                                                                  F                        2                                            /                                              P                        2                                                                                                                                                        F                      2                                        /                                          P                      2                                                        -                                                            [                                                                                                    (                                                                                          F                                2                                                            /                                                              P                                2                                                                                      )                                                                                low                            ⁢                                                          xe2x80x83                                                        ⁢                            pH                                                                                                                                (                                                                                          F                                1                                                            /                                                              P                                1                                                                                      )                                                                                low                            ⁢                                                          xe2x80x83                                                        ⁢                            pH                                                                                              ]                                        ⁢                                                                  F                        1                                            /                                              P                        1                                                                                                        )                        .                              
Since the present invention determines analyte content in a medium as a function of hydrogen ion concentration near the dye material:       Y    analyte    =            f      ⁢              (                                            K              a                        ⁡                          [                                                                    (                                                                  F                        2                                            /                                              P                        2                                                              )                                                        high                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                                                        (                                                                  F                        1                                            /                                              P                        1                                                              )                                                        low                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                              ]                                ⁢                                                                      F                  1                                /                                  P                  1                                            -                                                [                                                                                    (                                                                              F                            1                                                    /                                                      P                            1                                                                          )                                                                    high                        ⁢                                                  xe2x80x83                                                ⁢                        pH                                                                                                            (                                                                              F                            2                                                    /                                                      P                            2                                                                          )                                                                    high                        ⁢                                                  xe2x80x83                                                ⁢                        pH                                                                              ]                                ⁢                                                      F                    2                                    /                                      P                    2                                                                                                                        F                  2                                /                                  P                  2                                            -                                                [                                                                                    (                                                                              F                            2                                                    /                                                      P                            2                                                                          )                                                                    low                        ⁢                                                  xe2x80x83                                                ⁢                        pH                                                                                                            (                                                                              F                            1                                                    /                                                      P                            1                                                                          )                                                                    low                        ⁢                                                  xe2x80x83                                                ⁢                        pH                                                                              ]                                ⁢                                                      F                    1                                    /                                      P                    1                                                                                      )              .  
In the case of a sensor using optical absorption measurements to determine analyte content, the first and second illumination wavelengths are most preferably the peak absorption wavelengths of the undissociated dye molecules and their conjugate base ions, respectively. In the case of a sensor using fluorescence emission measurements to determine analyte content, the dye solution is most preferably illuminated by the peak excitation frequencies of the undissociated dye molecule and its conjugate base. Illumination of the dye material at these frequencies likely will maximize the intensity of the light returning from the dye material which, in turn, likely will increase the resolution.
In the case of a pH sensor, pHxe2x89xa1xe2x88x92log[H+]. Thus, the following modifications of the Henderson-Hasselbach equation relate pH values in a medium of interest to absorbance and to normalized fluorescence emission measurements obtained in accordance with the present invention:             pH      =                        pK          a                +                  log          ⁢                      xe2x80x83                    ⁢                      (                                          [                                                                            (                                              A                        1                                            )                                                              low                      ⁢                                              xe2x80x83                                            ⁢                      pH                                                                                                  (                                              A                        2                                            )                                                              high                      ⁢                                              xe2x80x83                                            ⁢                      pH                                                                      ]                            ⁢                              xe2x80x83                            ⁢                                                                    A                    2                                    -                                                                                                              (                                                      A                            2                                                    )                                                                          low                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                                                      (                                                      A                            1                                                    )                                                                          low                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                      ⁢                                          A                      1                                                                                                            A                    1                                    -                                                                                                              (                                                      A                            1                                                    )                                                                          high                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                                                      (                                                      A                            2                                                    )                                                                          high                          ⁢                                                      xe2x80x83                                                    ⁢                          pH                                                                                      ⁢                                          A                      2                                                                                            )                                ,          
        ⁢          and      ⁢              :                  pH    =                  pK        a            +                        log          ⁡                      (                                          [                                                                            (                                                                        F                          1                                                /                                                  P                          1                                                                    )                                                              low                      ⁢                                              xe2x80x83                                            ⁢                      pH                                                                                                  (                                                                        F                          2                                                /                                                  P                          2                                                                    )                                                              high                      ⁢                                              xe2x80x83                                            ⁢                      pH                                                                      ]                            ⁢                              xe2x80x83                            ⁢                                                                                          F                      2                                        /                                          P                      2                                                        -                                                            [                                                                                                    (                                                                                          F                                2                                                            /                                                              P                                2                                                                                      )                                                                                low                            ⁢                                                          xe2x80x83                                                        ⁢                            pH                                                                                                                                (                                                                                          F                                1                                                            /                                                              P                                1                                                                                      )                                                                                low                            ⁢                                                          xe2x80x83                                                        ⁢                            pH                                                                                              ]                                        ⁢                                                                  F                        1                                            /                                              P                        1                                                                                                                                                        F                      1                                        /                                          P                      1                                                        -                                                            [                                                                                                    (                                                                                          F                                1                                                            /                                                              P                                1                                                                                      )                                                                                high                            ⁢                                                          xe2x80x83                                                        ⁢                            pH                                                                                                                                (                                                                                          F                                2                                                            /                                                              P                                2                                                                                      )                                                                                high                            ⁢                                                          xe2x80x83                                                        ⁢                            pH                                                                                              ]                                        ⁢                                                                  F                        2                                            /                                              P                        2                                                                                                                  )                          .            
where pKaxe2x89xa1xe2x88x92log Ka, xe2x80x9cKaxe2x80x9d being the dissociation constant of the dye.
The partial pressure of carbon dioxide can be determined from absorbances and normalized fluorescent emissions measured for a dye solution containing a weakly acidic dye and HCO3xe2x88x92as follows:                                                         K              a1                        ⁢                          K              H                                                          K              a                        ⁡                          [                              HCO                3                -                            ]                                      ⁢                  (                                                    (                                  A                  1                                )                                            low                ⁢                                  xe2x80x83                                ⁢                pH                                                                    (                                  A                  2                                )                                            high                ⁢                                  xe2x80x83                                ⁢                pH                                              )                ⁢                  xe2x80x83                ⁢                  P                      CO            2                              =                                    A            1                    -                                                                      (                                      A                    1                                    )                                                  high                  ⁢                                      xe2x80x83                                    ⁢                  pH                                                                              (                                      A                    2                                    )                                                  high                  ⁢                                      xe2x80x83                                    ⁢                  pH                                                      ⁢                          A              2                                                            A            2                    -                                                                      (                                      A                    2                                    )                                                  low                  ⁢                                      xe2x80x83                                    ⁢                  pH                                                                              (                                      A                    1                                    )                                                  low                  ⁢                                      xe2x80x83                                    ⁢                  pH                                                      ⁢                          A              1                                            ,          
        ⁢          and      ⁢              :                                                                    K              a1                        ⁢                          K              H                                                          K              a                        ⁡                          [                              HCO                3                -                            ]                                      ⁡                  [                                                    (                                                      F                    1                                    /                                      P                    1                                                  )                                            low                ⁢                                  xe2x80x83                                ⁢                pH                                                                    (                                                      F                    2                                    /                                      P                    2                                                  )                                            high                ⁢                                  xe2x80x83                                ⁢                pH                                              ]                    ⁢              xe2x80x83            ⁢              P                  CO          2                      =                                                      F              1                        /                          P              1                                -                                    [                                                                    (                                                                  F                        1                                            /                                              P                        1                                                              )                                                        high                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                                                        (                                                                  F                        2                                            /                                              P                        2                                                              )                                                        high                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                              ]                        ⁢                                          F                2                            /                              P                2                                                                                        F              2                        /                          P              2                                -                                    [                                                                    (                                                                  F                        2                                            /                                              P                        2                                                              )                                                        low                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                                                        (                                                                  F                        1                                            /                                              P                        1                                                              )                                                        low                    ⁢                                          xe2x80x83                                        ⁢                    pH                                                              ]                        ⁢                                          F                1                            /                              P                1                                                        .      
An especially preferred system for carrying out the invention includes an electronic controller comprising a programmed microprocessor in electrical communication with the light sources and the photodetector systems. Most preferably, the microprocessor is programmed to modulate the light sources; to accept signals representing intensity measurements from the photodetectors during time periods associated with the illumination of the dye by one or more wavelengths of light; and to process the intensity measurements in accordance with the complex ratiometric expressions given above. The preparation of a suitable program in accordance with the invention as disclosed herein is within the ordinary skill in the art and requires no undue experimentation.
Therefore, it is one object of the invention to provide methods and systems by which the analyte content of a medium can be determined to high resolution by exposing a dye solution to the analyte; measuring spectral properties of the dye corresponding to two wavelengths of light; and processing these spectral properties in accordance with a complex ratiometric expression. The invention will be further described in conjunction with the appended drawings and following detailed description.